‘Turnout in the Lewisham East byelection was miserably low: a mere third of eligible voters took part.’ The Labour candidate Janet Daby celebrates with her supporters
Photograph: Yui Mok/PA

Labour’s resounding victory in Lewisham East is destined to be interpreted thus: because the Liberal Democrats surged from a paltry 4.4% to nearly a quarter of the vote, while Labour’s share fell from two-thirds to just over 50%, Labour should pivot to a firmly anti-Brexit stance.

For those of us who don’t like Brexit but particularly fear an ideologically driven Tory hard Brexit, this would be a big mistake.

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Turnout in the byelection was miserably low: a mere third of eligible voters took part. In last year’s general election, nearly seven out of 10 participated, handing Labour its biggest ever victory in the constituency, surpassing the figure chalked up in the landslide year of 1997 by nearly 10 percentage points. Both turnout and Labour’s success are linked. Corbyn’s Labour mobilised younger voters and previous non-voters who may be persuaded to march to a polling station in a general election, but not in a byelection. Those particularly fired up about an issue – in this case Brexit – can make a particular impact in a low-turnout byelection, knowing that they won’t risk a Tory victory in doing so.

Lewisham East is one of the top 10% most pro-remain constituencies in Britain – and yet, even with Brexit in acute crisis, the avowedly pro-remain Lib Dems got a lower share of the vote than in 2010. That they can celebrate this result is a sign of how far the Lib Dems have plummeted – and in another high turnout general election, it would seem reasonable to expect a result similar to last year’s.

Brexit is a Tory-engineered mess that Labour did not want. The party has to walk a political tightrope on Brexit because it represents staunchly pro-remain communities like Lewisham, Hackney and Islington, as well as heavily pro-leave communities like Doncaster, Hull and Ashfield. After Labour accepted the referendum result, voted for the triggering of article 50, and made no firm commitments on the single market or customs union in the general election, the narrative was that it would fatally alienate remain voters. But that did not happen. Since then it has softened its approach to Brexit, not least committing to a customs union – and it rightly frets over its leave flank. The only seats Labour lost last year voted decisively for Brexit, such as Mansfield, Middlesbrough South and East Cleveland, and Stoke-on-Trent South.

Labour’s majorities slumped in strongly pro-leave constituencies (such as Ashfield, Dudley North and Bolsover), and Labour MPs representing these communities report that Brexit supporters ask in baffled tones why Britain hasn’t left the EU yet – and believe politicians are trying to thwart their vote. The reason those such as Gloria De Piero and Caroline Flint – Labour MPs who could hardly be described as Corbynites – are emphatic about the need to implement the referendum result is because they know what their constituents are telling them. That doesn’t mean signing up to whatever the Tories put forward, no matter how hard a Brexit, but it does mean accepting that the debate is about how we leave the EU, not if we should.

Amid all the noise, the fact is if you voted leave or remain in 2016, polling suggests you would vote the same way again. To borrow Theresa May’s phrase, nothing has changed: not meaningfully, anyway. There is a strong case to be made that, after a second referendum campaign, Brexit would win an even bigger majority, poison our politics even more, and make the likelihood of a hard Brexit all the more likely. There are remain voters who believe the referendum result should be implemented; there are others who would like it reversed, but will vote Labour because they understand what the alternative is.

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If Labour is to form a majority government with a transformative programme to deal with some of the injustices that drove Brexit in the first place, as well as avoid an ideologically driven Tory Brexit, it needs to build a broad coalition of remain and leave voters. That means offering a compromise: accepting the referendum result but remaining in a customs union, for example.

It must also talk about other issues. May wanted to frame last year’s general election as a rerun of the referendum. Labour did not let her, shifting the conversation on other domestic issues, allowing it to deprive the Tories of their majority. The next election is not in the bag for Labour, and its leaders know that. Outside of an election campaign, they haven’t managed to recapture the sense of insurgency that drove their vote surge last year, and that should worry them. But if they fall off their Brexit tightrope, Jacob Rees-Moggified Tories will win the next general election, leaving us with the worst Brexit of all and a profoundly reactionary government to boot.